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Full text of "Mathematical And Physical Papers - Iii"

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;                           This equation shews that if we represent the fluxes across
planes perpendicular to the axes of x, y, z, by three forces or three velocities, the flux across any other plane will be repre-
:                      sented by the resolved part of the forces or velocities along the
normal to this plane. Hence the flux across one particular plane passing through P is a maximum, and the flux across any other plane is equal to this maximum flux multiplied by the cosine of the angle between the two planes.
2.    Let u be the temperature at P at the end of the time t, and consider the portion of the solid which is contained in the elementary volume  dxdydz adjacent  to  P.    The  quantity  of ,                              heat which enters this element during the time dt by the first
lj                              of the faces dy dz is  ultimately equal, to fx dy dz dt, and  the
!f>                                 quantity which escapes by the opposite face is ultimately equal
'*                                 to (fx + dfxjdoo. dx) dy dz dt.    Subtracting  the  former from  the
'',jl                                latter, and treating in the same way each of the other two pairs
%\                              of opposite faces, we find that the loss of heat in the element is
l\i ]                              ultimately equal to
++   dxdl/dsdt
dy     dz I
But if p be the density, p dxdydz will be the mass; and if c be the specific heat, the loss of heat will also be equal to
 p dx dy dz . c -?- dt
ultimately.    Equating the two results, and passing to the limit, we get
3. The formulae (1) and (2) are general, but for the future I shall suppose the medium to be homogeneous, and the temperature to differ by only a small quantity from a certain fixed standard which we may suppose to be the origin from which u is measured. Since the medium is homogeneous p is constant*, and c moreover will be constant, except so far as relates to a change of specific heat produced by a change of temperature. But since u is supposed to be small, the terms arising from the
* The expansion of the solid produced by heat is not here taken into account.